Question

The point $(4\cos \theta ,4\sin \theta )$ lies ____________ the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$

• A. outside
• B. inside
• C. on the periphery
• D. on the auxiliary circle

Answer 1

The correct option is D on the auxiliary circle

Equation of ellipse is $\frac{x^2}{16}+\frac{y^2}{9}=1$

Equation of auxiliary circle of ellipse is

$x^2+y^2=a^2{x}^2+y^2=16$

Substiuting $(4\cos \theta ,4\sin \theta$) in the equation, we get

${\left(4\cos \theta \right)}^2+{\left(4\sin \theta \right)}^2=1616({\cos }^2\theta +{\sin }^2\theta )=1616=16$

The point satisfies the equation of auxiliary circle.

So, option D is the correct.